nLab commutative monoid in a symmetric monoidal model category

Idea

Let $C$ be symmetric monoidal category and $C$ the category of commutative monoids in $C$. When $C$ is further a model category, there are certain conditions under which there is an induced model structure on $CMon(C)$, where the weak equivalences and fibrations are defined as in $C$.

Distinction between E-infinity monoids

Recall that $CMon(C)$ can be described as the category of algebras over an operad over the operad Comm. If the operad Comm were cofibrant, then for the existence of the induced model structure on $CMon(C)$ it would be sufficient to require the monoid axiom, and to use the model structure on algebras over an operad as discussed there. However Comm is in general not cofibrant, and this is the distinction between Comm and the E-infinity operad: the latter is a cofibrant replacement of the former. This is why the model structure on commutative monoids may not always exist even when the model structure on E-infinity monoids in a symmetric monoidal model category does.

For example, take $C$ to be the category of chain complexes over a field of positive characteristic. Then the category of commutative monoids in $C$ is the category of commutative dg-algebras. This does not have an induced model structure, as explained in MO/23885/2503.

See Rectification for some results on when the model structures on E-infinity monoids? and commutative monoids are Quillen equivalent, though.

Model structure

Let $C$ be a symmetric monoidal model category and let $CMon(C)$ denote the category of commutative monoids in the underlying category of $C$. Define a weak equivalence (resp. fibration) of commutative monoids to be a weak equivalence (resp. fibration) of the underlying objects of $C$. Below we will give sufficient conditions for this to define a model structure on $CMon(C)$.

Theorem

Suppose that

• $C$ is combinatorial,

• $C$ is freely powered, i.e.

  • $C$ satisfies the monoid axiom in a monoidal model category,

  • $C$ is left proper and …

  • every cofibration in $C$ is a power cofibration, i.e. …

Then the category $CMon(C)$ is a combinatorial model category with weak equivalences and fibrations as defined above.

See (Lurie, Proposition 4.5.4.6).

Next we state a more general version of this result, for which we will require some set-theoretic assumptions. Suppose that $C$ is cofibrantly generated by a set of cofibrations $I$ and a set of trivial cofibrations $J$. Let $I \otimes C$-cell denote the closure of $I \otimes C = \{ i \otimes \id_X : i \in I, X \in Ob(C) \}$ under cobase change and transfinite composition, and similarly for $J \otimes C$-cell.

Theorem

Suppose that

• $C$ is cofibrantly generated by a set of cofibrations $I$ and a set of fibrations $J$,

• the domains of the morphisms in $I$ (resp. $J$) are small objects relative to $(I\otimes C)$-cell (resp. $(J\otimes C)$-cell),

• $C$ satisfies the monoid axiom in a monoidal model category

• $C$ satisfies the commutative monoid axiom, i.e. …

Then the category $CMon(C)$ is a cofibrantly generated model category with weak equivalences and fibrations as defined above.

If $C$ is further simplicial (resp. combinatorial, tractable), then so is $CMon(C)$.

See (White 14, Theorem 3.2).

Rectification

Rectification of $E_\infty$-monoids is the question of whether the weak equivalence between the operads Comm and the E-infinity operad induces a Quillen equivalence on the model categories of algebras. Since the model category of algebras over an operad over the E-infinity operad is a presentation of the (infinity,1)-category of commutative monoids in a symmetric monoidal (infinity,1)-category, rectification for $C$ is equivalent to saying that the (infinity,1)-category presented by the model structure on commutative monoids is equivalent to the (infinity,1)-category of commutative monoids in a symmetric monoidal (infinity,1)-category in the symmetric monoidal (infinity,1)-category presented by $C$.

The following cases are particularly interesting.

• Rectification holds in the model category of symmetric spectra.
• It does not hold in the model category of simplicial sets or in the model category of simplicial abelian groups.

See (Lurie, Theorem 4.5.4.7) for sufficient conditions for rectification to hold. See also (White 14, Paragraph 4.2) for more discussion.

A general rectification criterion for symmetric monoidal model categories is formulated in PS 14, Proposition 10.1.2 and Theorem 9.3.6. It says that given a tractable? symmetric monoidal model category that satisfies a certain compact generatedness assumption with a morphism of admissible operads? $A\to B$ (e.g., $A=E_\infty$, $B=Comm$), the Quillen adjunction between $A$-monoids and $B$-monoids induced by the morphism of operads $A\to B$ is a Quillen equivalence if and only if for any cofibration $s$ and any $n\ge0$ the morphism $(A_n\to B_n)\wedge_{\Sigma_n}s^{\wedge n}$ is a weak equivalence, where $\wedge$ denotes the pushout product with respect to the monoidal structure.

Related concepts

• model structure on E-infinity monoids in a symmetric monoidal model category

• model structure on monoids in a monoidal model category

• commutative monoid in a symmetric monoidal (infinity,1)-category

• model structure on algebras over an operad

• model structure on algebras over a monad

• model structure on simplicial T-algebras

References

• David White, Model Structures on Commutative Monoids in General Model Categories, arXiv:1403.6759.

• Jacob Lurie, Higher Algebra.

• Dmitri Pavlov and Jakob Scholbach, Admissibility and rectification of colored symmetric operads, arXiv:1410.5675.